Why does it matter who pulls the permit?
You want the Contractor’s to pull the permits. If the contractor pulls the permits they will be required to provide licenses and insurance and will be held to higher standards. Also if they don’t complete the project or do shoddy work you will have help to enforce rules and regs.
It is difficult to say why a contractor would push the owner to pull the permit. One reason is possibly to save his client money. Or to save himself from offering this as a part of his service that he did not factor charging for.
As a home owner (project owner hiring a contractor) it is in your best interest to have this as a part of the service the contractor provides. In most jurisdictions that our company is familiar with the one responsible for upholding the law (building codes) is the party that signs for and picks up the permit.
If you decide to save money by doing this service yourself, for your contractor, it could result in getting burned so to speak. If the contractor fails to build according to the code and the approved plans, you as the one who signed for and received the permit can become responsible. This could end up costing you thousands if not tens of thousands more.
If you encounter a “Contractor” asking for you as the homeowner to pull the permit that is usually a Red Flag that something is not right. They usually are not a Licensed Contractor. There is no savings in cost for a homeowner to pull a permit.
Secondly if you as the Homeowner pull a permit you are fully responsible and liable for the work. Technically if you still choose to hire them to work under your permit you are required to carry workers compensation and liability insurance on them as they become your employee.
As a small electrical contracting business (fewer than 20 employees) we simply don’t always have the manpower to have an employee go to pull the permit. We will pull the permit if the customer insists, but we charge an hourly rate plus the cost of the permit. In our case, it is cheaper for the customer to pull the permit themselves. We do let our customers know upfront that permit costs are excluded, and have been operating our business this way for over 30 years with no problems.
Most likely they are not a Licensed Contractor!!! Or been flagged by CSLB for unfair business deals etc…
Happens quite often unfortunately. 
My answer is similar to the answer Country Bear Electric Inc provided. If a contractor is providing you with the proper paperwork (Insurance/liability, workers comp, license, etc.) and gives you the option to pull the permits yourself using the companies information, then they are simply lowering their operating costs by not traveling to the city, spending x amount of hours waiting to submit the permits, returning to the city, waiting x amount of hours to pick up the permit, and any other city related work they need to complete. AAGC avoids having owners submit their own paper work, but if a homeowner asks for a discount we usually offer a discount if the owner does the busy work of handling the permits.